I get what was said above that you're searching for advice on loneliness, but I can't avoid saying, I don't think this is sustainable. Being in the closet about something so central to your life is just painful. Is there any possibility of moving, if not now, maybe a couple of years down the line? You deserve more. Its not about public displays of affection, some people who are married don't even do that, its about not having to lie and pretend.

As for the loneliness... maybe you could ask their permission to tell one or two very close, trusted friends? Maybe you could join a poly social group in the nearest city and all go there once a month?